How Do You Convert Frequency Response Function H(iw) to Time Domain h(t)?

[magnifik]

I am trying to find the frequency response function given the input and output spectra

from the picture i got that X(iw) = iU(w)U(2-w) - iU(w+2)U(w) and that
Y(iw) = iwU(w)U(1-w) - iwU(w+1)U(w)

So H(iw) = Y(iw)/X(iw)
and H(iw) = wU(1-w) - wU(w+1) / U(2-w) - U(w+2)

i am having trouble now converting H(iw) to h(t) as i have never divided unit step functions before. can someone tell me if my work above is correct and if so, how to go about finding h(t)? thanks.

 

[KataKoniK]

Your work appears to be correct so far. To convert H(iw) to h(t), you can use the inverse Fourier transform. This will give you the time domain representation of the frequency response function. The steps to do this are as follows:

1. Rewrite H(iw) in the form of a Fourier transform pair, where H(iw) is the transform of h(t). In this case, H(iw) = wU(1-w) - wU(w+1) / U(2-w) - U(w+2) can be rewritten as H(iw) = wU(1-w) - wU(w+1) / (2/pi) * [delta(w-2) + delta(w+2)].

2. Take the inverse Fourier transform of H(iw) to get h(t). This can be done by using the properties of the delta function and the Fourier transform. The result will be h(t) = 1/2 * [tU(t) - (t-2)U(t-2) - (t+2)U(t+2)].

3. Simplify the expression for h(t) to get the final form of the frequency response function. In this case, h(t) can be simplified to h(t) = 1/2 * [tU(t) - tU(t-2) - tU(t+2) + 2U(t+2)].

I hope this helps. Let me know if you have any further questions. Good luck with your work!